170 XIII INTEGRATION
converges. Note that if m2 g N < (m + I)2, we have
, . m2
'<*)--/.ğ<>
and deduce that the sequence (/N(X))NğI tends to 0 for almost all x e [0,1] (with
respect to Lebesgue measure). Hence show that, for almost all x e [0,1], the sequence
(An* [An;t]) is equirepartitioned with respect to Lebesgue measure (Section 13.4,
Problem 7).
16. Let U be a continuous endomorphism of the space LR(X, p) with norm <^ 1 (in other
words, such that NI(£/ /) <£ NI(/)) and such that the relation/^ 0 implies U-f^O
(13.6). If/is any function in the class f, we denote by U /any function in the class
u-f.
(a) Let (fn) be a sequence of functions in Ğ^R which converges almost everywhere
to a function/and is such that there exists a function h e 3?^ satisfying \fn\ ^ /z for
all n. Show that under these conditions the sequence (U-fn) converges almost every-
where to £/*/ (Reduce to the case /= 0, and consider the sequence of functions
(b) For each finite sequence (tk)i;^k^n of real numbers, put
, ...,/) >0, then
S-n('l,...sO = SUP (*i + -
l<fc<n
Show that if sn(tlf . . . , tn) > 0, then for all /B+1 6 R we have
(1) *i + sn(t2 , . . . , tn+1) ^ 5n(/i, . . . , tn).
(c) For each function / e ^R , show that
sJiU-f, . . . , £/n+1 /) < t/ - JH(/, ...,(/" /)
almost everywhere. (Use (1) and the fact that for any n functions fl9 ...,/ 6 &^
we have
sup U fk ^ U ' sup fk
l^k^n l^fc^n
almost everywhere.)
(d) Let E(/) be the measurable set of points x e X such that
/(*) + (U-f)(x) + - + (t/"-1 /)(*) > 0
for some integer n > 0. Show that
> 0
~-
f
JE(/)
(E. Hopf's maximal ergodic theorem). (For each integer n ^> 1, let En(/) denote the
set of x e X such that
/(*) + ((//)(*) + - + (£/* /)W > o